Lemma 0 (Corollary of Assumptions): ([1]} Lemma 2.2, [2]} Lemma 1) Under assumptions 1 and 2 stated above, for all \(x, x_1, x_2 \in \mathcal {X}, y \in \mathbb {R}^{d_2}, i \in [n]\) , we have \( \Vert \overline{\nabla }_x f_i(x,y)-\nabla \ell _i(x)\Vert  \le L\Vert y_i^*(x)-y\Vert \) , \(\Vert y_i^*(x_1)-y_i^*(x_2)\Vert  \le G_y \Vert x_1-x_2\Vert \) , and \(\Vert \nabla \ell _i(x_1)-\nabla \ell _i(x_2)\Vert  \le L_f\Vert x_1-x_2\Vert \) .
Here, we define
\( L &\triangleq L_{f_x} + \frac{L_{f_y} C_{gxy}}{\mu _g} + C_{f_y} \left(\frac{L_{gxy}}{\mu _g} + \frac{L_{gyy} C_{gxy}}{\mu _g^2}\right),\\ L_f &\triangleq L_{f_x} + \frac{(\bar{L}_{f_y} + L) C_{gxy}}{\mu _g} + C_{f_y} \left( \frac{\bar{L}_{gxy}}{\mu _g} + \frac{\bar{L}_{gyy} C_{gxy}}{\mu _g^2}\right), G_y \triangleq \frac{C_g}{\mu _g}.\) 
